Optimal. Leaf size=301 \[ -\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}+1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}} \]
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Rubi [A] time = 0.60, antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2342, 266, 63, 208, 2348, 5984, 5918, 2402, 2315} \[ -\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {PolyLog}\left (2,-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}}+1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right ) \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 2315
Rule 2342
Rule 2348
Rule 2402
Rule 5918
Rule 5984
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {1-\frac {e^2 x^2}{d^2}}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{x} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}\left (\sqrt {1-\frac {e^2 x}{d^2}}\right )}{x} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \operatorname {Subst}\left (\int \frac {x \tanh ^{-1}(x)}{-1+x^2} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{1-x} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,\sqrt {1-\frac {e^2 x^2}{d^2}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (b n \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right )^2}{2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \tanh ^{-1}\left (\sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \log \left (\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {b n \sqrt {1-\frac {e^2 x^2}{d^2}} \text {Li}_2\left (1-\frac {2}{1-\sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 1.83, size = 310, normalized size = 1.03 \[ -\frac {\log \left (\sqrt {d-e x} \sqrt {d+e x}+d\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d}+\frac {\log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d}+\frac {b n \sqrt {e^2 x^2-d^2} \left (\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {e^2 x^2}{d^2}}\right )+\log ^2\left (\frac {e^2 x^2}{d^2}\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {1-\frac {e^2 x^2}{d^2}}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {1-\frac {e^2 x^2}{d^2}}+1\right )\right ) \log \left (\frac {e^2 x^2}{d^2}\right )\right )}{\sqrt {e^2 x^2-d^2}}-\frac {4 \left (2 \log (x)-\log \left (\frac {e^2 x^2}{d^2}\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e^2 x^2-d^2}}{\sqrt {-d^2}}\right )}{\sqrt {-d^2}}\right )}{8 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {-e x + d} b \log \left (c x^{n}\right ) + \sqrt {e x + d} \sqrt {-e x + d} a}{e^{2} x^{3} - d^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x + d} \sqrt {-e x + d} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\sqrt {-e x +d}\, \sqrt {e x +d}\, x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{\sqrt {e x + d} \sqrt {-e x + d} x}\,{d x} - \frac {a \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\sqrt {d+e\,x}\,\sqrt {d-e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{x \sqrt {d - e x} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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